FA-70126 / Map projection transforms / Open access

Spherical Albers equal-area conic forward: cone constant averaging · case 01

Area totals drift and meridians converge twice as fast as intended.

Verified by executionVariant 1 · 8 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The cone constant sums the sines of the standard parallels without halving.

VERIFIED REPAIR

At the cone constant averaging step restore `n = (math.sin(p1) + math.sin(p2)) / 2`, leaving the rest of the model unchanged.

Unsuccessful approach: Taking the sine of the mean parallel matches only tangent cones.

Case contract

Input [lon, lat, lon0, lat0, lat1, lat2] degrees, sphere R = 6371000, lat1 != -lat2. n = (sin p1 + sin p2)/2; C = cos(p1)**2 + 2*n*sin(p1); rho = R*sqrt(C - 2*n*sin(phi))/n; rho0 likewise at lat0; theta = n*dlon with dlon wrapped to [-180, 180). x = rho*sin(theta), y = rho0 - rho*cos(theta), rounded to 2 decimals.

Why this case matters

Continental statistics maps use Albers because it preserves area; a wrong constant silently biases area totals.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    lon, lat, lon0, lat0, lat1, lat2 = x
    R = 6371000.0
    p0, p1, p2, phi = [math.radians(v) for v in (lat0, lat1, lat2, lat)]
    n = math.sin(p1) + math.sin(p2)
    C = math.cos(p1) ** 2 + 2 * n * math.sin(p1)
    rho = R * math.sqrt(C - 2 * n * math.sin(phi)) / n
    rho0 = R * math.sqrt(C - 2 * n * math.sin(p0)) / n
    dl = math.radians(((lon - lon0 + 180.0) % 360.0) - 180.0)
    theta = n * dl
    return [round(rho * math.sin(theta), 2), round(rho0 - rho * math.cos(theta), 2)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('control #0', [-74.411, 16.812, -96.0, 23.0, 33.0, 45.0], [2405679.11, -375515.8]), ('control #1', [-7.054, -2.555, 10.0, 52.0, 35.0, 65.0], [-2368752.85, -5460256.95]), ('control #2', [163.654, -41.81, 132.0, 0.0, -18.0, -36.0], [2661851.12, -4920407.16]), ('control #3', [-52.492, -45.412, -60.0, -32.0, -5.0, -42.0], [601606.86, -1530950.07]), ('control #4', [-26.25, -1.472, 0.0, 40.0, 40.0, 40.0], [-3459655.33, -3813601.11]), ('control #5', [24.242, -15.571, 20.0, -45.0, -45.0, -45.0], [499045.1, 3152788.69]), ('boundary #28', [-96.0, 23.0, -96.0, 23.0, 33.0, 45.0], [0.0, 0.0]), ('boundary #29', [5.0, 40.0, 5.0, 40.0, 40.0, 40.0], [0.0, 0.0])], [('control #1', [-7.054, -2.555, 10.0, 52.0, 35.0, 65.0], [-2368752.85, -5460256.95]), ('control #2', [163.654, -41.81, 132.0, 0.0, -18.0, -36.0], [2661851.12, -4920407.16]), ('control #5', [24.242, -15.571, 20.0, -45.0, -45.0, -45.0], [499045.1, 3152788.69]), ('control #7', [-119.961, 31.565, -96.0, 23.0, 33.0, 45.0], [-2250314.51, 1235327.67]), ('control #8', [-25.931, 8.989, 10.0, 52.0, 35.0, 65.0], [-4374379.83, -3619965.56]), ('control #10', [-40.992, -24.699, -60.0, -32.0, -5.0, -42.0], [1816236.48, 739740.04]), ('boundary #28', [-96.0, 23.0, -96.0, 23.0, 33.0, 45.0], [0.0, 0.0]), ('boundary #29', [5.0, 40.0, 5.0, 40.0, 40.0, 40.0], [0.0, 0.0])], [('control #2', [163.654, -41.81, 132.0, 0.0, -18.0, -36.0], [2661851.12, -4920407.16]), ('control #3', [-52.492, -45.412, -60.0, -32.0, -5.0, -42.0], [601606.86, -1530950.07]), ('control #10', [-40.992, -24.699, -60.0, -32.0, -5.0, -42.0], [1816236.48, 739740.04]), ('control #11', [25.244, 44.637, 0.0, 40.0, 40.0, 40.0], [1977746.83, 796959.72]), ('control #12', [-19.298, -12.334, 20.0, -45.0, -45.0, -45.0], [-4597338.48, 2353100.16]), ('control #14', [-128.991, 27.029, -96.0, 23.0, 33.0, 45.0], [-3243071.92, 1029568.66]), ('boundary #28', [-96.0, 23.0, -96.0, 23.0, 33.0, 45.0], [0.0, 0.0]), ('boundary #29', [5.0, 40.0, 5.0, 40.0, 40.0, 40.0], [0.0, 0.0])], [('control #3', [-52.492, -45.412, -60.0, -32.0, -5.0, -42.0], [601606.86, -1530950.07]), ('control #7', [-119.961, 31.565, -96.0, 23.0, 33.0, 45.0], [-2250314.51, 1235327.67]), ('control #14', [-128.991, 27.029, -96.0, 23.0, 33.0, 45.0], [-3243071.92, 1029568.66]), ('control #15', [-24.721, 3.031, 10.0, 52.0, 35.0, 65.0], [-4481724.64, -4195384.79]), ('control #17', [-37.667, -43.269, -60.0, -32.0, -5.0, -42.0], [1818158.31, -1416145.5]), ('control #19', [-7.283, -57.541, 20.0, -45.0, -45.0, -45.0], [-1648585.12, -1661270.39]), ('boundary #28', [-96.0, 23.0, -96.0, 23.0, 33.0, 45.0], [0.0, 0.0]), ('boundary #29', [5.0, 40.0, 5.0, 40.0, 40.0, 40.0], [0.0, 0.0])], [('control #4', [-26.25, -1.472, 0.0, 40.0, 40.0, 40.0], [-3459655.33, -3813601.11]), ('control #7', [-119.961, 31.565, -96.0, 23.0, 33.0, 45.0], [-2250314.51, 1235327.67]), ('control #19', [-7.283, -57.541, 20.0, -45.0, -45.0, -45.0], [-1648585.12, -1661270.39]), ('control #21', [-59.567, 18.585, -96.0, 23.0, 33.0, 45.0], [3917697.15, 316916.45]), ('control #22', [40.214, 4.407, 10.0, 52.0, 35.0, 65.0], [3883538.77, -4321985.71]), ('control #24', [-22.139, -46.424, -60.0, -32.0, -5.0, -42.0], [2976538.67, -1998953.15]), ('boundary #28', [-96.0, 23.0, -96.0, 23.0, 33.0, 45.0], [0.0, 0.0]), ('boundary #29', [5.0, 40.0, 5.0, 40.0, 40.0, 40.0], [0.0, 0.0])]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
control #0	`[2679761.39, 56664.72]`	`[2405679.11, -375515.8]`	Failed
control #1	`[-2902670.03, -5337287.77]`	`[-2368752.85, -5460256.95]`	Failed
control #2	`[1732222.4, -5374949.58]`	`[2661851.12, -4920407.16]`	Failed
control #3	`[180733.24, -2967656.07]`	`[601606.86, -1530950.07]`	Failed
control #4	`[-4180203.63, -2460648.73]`	`[-3459655.33, -3813601.11]`	Failed
control #5	`[621199.55, 2725718.09]`	`[499045.1, 3152788.69]`	Failed
boundary #28	`[0.0, 0.0]`	`[0.0, 0.0]`	Passed
boundary #29	`[0.0, 0.0]`	`[0.0, 0.0]`	Passed

SHA-256 / 217c4e0133c776225bb64b5b34d0e6dbfc3b7d55e7c1e7535f97d8d0ed42d972

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    lon, lat, lon0, lat0, lat1, lat2 = x
    R = 6371000.0
    p0, p1, p2, phi = [math.radians(v) for v in (lat0, lat1, lat2, lat)]
    n = math.sin((p1 + p2) / 2)
    C = math.cos(p1) ** 2 + 2 * n * math.sin(p1)
    rho = R * math.sqrt(C - 2 * n * math.sin(phi)) / n
    rho0 = R * math.sqrt(C - 2 * n * math.sin(p0)) / n
    dl = math.radians(((lon - lon0 + 180.0) % 360.0) - 180.0)
    theta = n * dl
    return [round(rho * math.sin(theta), 2), round(rho0 - rho * math.cos(theta), 2)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('control #0', [-74.411, 16.812, -96.0, 23.0, 33.0, 45.0], [2405679.11, -375515.8]), ('control #1', [-7.054, -2.555, 10.0, 52.0, 35.0, 65.0], [-2368752.85, -5460256.95]), ('control #2', [163.654, -41.81, 132.0, 0.0, -18.0, -36.0], [2661851.12, -4920407.16]), ('control #3', [-52.492, -45.412, -60.0, -32.0, -5.0, -42.0], [601606.86, -1530950.07]), ('control #4', [-26.25, -1.472, 0.0, 40.0, 40.0, 40.0], [-3459655.33, -3813601.11]), ('control #5', [24.242, -15.571, 20.0, -45.0, -45.0, -45.0], [499045.1, 3152788.69]), ('boundary #28', [-96.0, 23.0, -96.0, 23.0, 33.0, 45.0], [0.0, 0.0]), ('boundary #29', [5.0, 40.0, 5.0, 40.0, 40.0, 40.0], [0.0, 0.0])], [('control #1', [-7.054, -2.555, 10.0, 52.0, 35.0, 65.0], [-2368752.85, -5460256.95]), ('control #2', [163.654, -41.81, 132.0, 0.0, -18.0, -36.0], [2661851.12, -4920407.16]), ('control #5', [24.242, -15.571, 20.0, -45.0, -45.0, -45.0], [499045.1, 3152788.69]), ('control #7', [-119.961, 31.565, -96.0, 23.0, 33.0, 45.0], [-2250314.51, 1235327.67]), ('control #8', [-25.931, 8.989, 10.0, 52.0, 35.0, 65.0], [-4374379.83, -3619965.56]), ('control #10', [-40.992, -24.699, -60.0, -32.0, -5.0, -42.0], [1816236.48, 739740.04]), ('boundary #28', [-96.0, 23.0, -96.0, 23.0, 33.0, 45.0], [0.0, 0.0]), ('boundary #29', [5.0, 40.0, 5.0, 40.0, 40.0, 40.0], [0.0, 0.0])], [('control #2', [163.654, -41.81, 132.0, 0.0, -18.0, -36.0], [2661851.12, -4920407.16]), ('control #3', [-52.492, -45.412, -60.0, -32.0, -5.0, -42.0], [601606.86, -1530950.07]), ('control #10', [-40.992, -24.699, -60.0, -32.0, -5.0, -42.0], [1816236.48, 739740.04]), ('control #11', [25.244, 44.637, 0.0, 40.0, 40.0, 40.0], [1977746.83, 796959.72]), ('control #12', [-19.298, -12.334, 20.0, -45.0, -45.0, -45.0], [-4597338.48, 2353100.16]), ('control #14', [-128.991, 27.029, -96.0, 23.0, 33.0, 45.0], [-3243071.92, 1029568.66]), ('boundary #28', [-96.0, 23.0, -96.0, 23.0, 33.0, 45.0], [0.0, 0.0]), ('boundary #29', [5.0, 40.0, 5.0, 40.0, 40.0, 40.0], [0.0, 0.0])], [('control #3', [-52.492, -45.412, -60.0, -32.0, -5.0, -42.0], [601606.86, -1530950.07]), ('control #7', [-119.961, 31.565, -96.0, 23.0, 33.0, 45.0], [-2250314.51, 1235327.67]), ('control #14', [-128.991, 27.029, -96.0, 23.0, 33.0, 45.0], [-3243071.92, 1029568.66]), ('control #15', [-24.721, 3.031, 10.0, 52.0, 35.0, 65.0], [-4481724.64, -4195384.79]), ('control #17', [-37.667, -43.269, -60.0, -32.0, -5.0, -42.0], [1818158.31, -1416145.5]), ('control #19', [-7.283, -57.541, 20.0, -45.0, -45.0, -45.0], [-1648585.12, -1661270.39]), ('boundary #28', [-96.0, 23.0, -96.0, 23.0, 33.0, 45.0], [0.0, 0.0]), ('boundary #29', [5.0, 40.0, 5.0, 40.0, 40.0, 40.0], [0.0, 0.0])], [('control #4', [-26.25, -1.472, 0.0, 40.0, 40.0, 40.0], [-3459655.33, -3813601.11]), ('control #7', [-119.961, 31.565, -96.0, 23.0, 33.0, 45.0], [-2250314.51, 1235327.67]), ('control #19', [-7.283, -57.541, 20.0, -45.0, -45.0, -45.0], [-1648585.12, -1661270.39]), ('control #21', [-59.567, 18.585, -96.0, 23.0, 33.0, 45.0], [3917697.15, 316916.45]), ('control #22', [40.214, 4.407, 10.0, 52.0, 35.0, 65.0], [3883538.77, -4321985.71]), ('control #24', [-22.139, -46.424, -60.0, -32.0, -5.0, -42.0], [2976538.67, -1998953.15]), ('boundary #28', [-96.0, 23.0, -96.0, 23.0, 33.0, 45.0], [0.0, 0.0]), ('boundary #29', [5.0, 40.0, 5.0, 40.0, 40.0, 40.0], [0.0, 0.0])]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
control #0	`[2407501.22, -373232.0]`	`[2405679.11, -375515.8]`	Failed
control #1	`[-2391339.96, -5438305.25]`	`[-2368752.85, -5460256.95]`	Failed
control #2	`[2652037.07, -4925906.76]`	`[2661851.12, -4920407.16]`	Failed
control #3	`[586482.05, -1561057.47]`	`[601606.86, -1530950.07]`	Failed
control #4	`[-3459655.33, -3813601.11]`	`[-3459655.33, -3813601.11]`	Passed
control #5	`[499045.1, 3152788.69]`	`[499045.1, 3152788.69]`	Passed
boundary #28	`[0.0, 0.0]`	`[0.0, 0.0]`	Passed
boundary #29	`[0.0, 0.0]`	`[0.0, 0.0]`	Passed

SHA-256 / 7df7004f8bbaa8f6991a2faa62f36651a97db40e7b35c6cf3b264ae60c58cfee

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    lon, lat, lon0, lat0, lat1, lat2 = x
    R = 6371000.0
    p0, p1, p2, phi = [math.radians(v) for v in (lat0, lat1, lat2, lat)]
    n = (math.sin(p1) + math.sin(p2)) / 2
    C = math.cos(p1) ** 2 + 2 * n * math.sin(p1)
    rho = R * math.sqrt(C - 2 * n * math.sin(phi)) / n
    rho0 = R * math.sqrt(C - 2 * n * math.sin(p0)) / n
    dl = math.radians(((lon - lon0 + 180.0) % 360.0) - 180.0)
    theta = n * dl
    return [round(rho * math.sin(theta), 2), round(rho0 - rho * math.cos(theta), 2)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('control #0', [-74.411, 16.812, -96.0, 23.0, 33.0, 45.0], [2405679.11, -375515.8]), ('control #1', [-7.054, -2.555, 10.0, 52.0, 35.0, 65.0], [-2368752.85, -5460256.95]), ('control #2', [163.654, -41.81, 132.0, 0.0, -18.0, -36.0], [2661851.12, -4920407.16]), ('control #3', [-52.492, -45.412, -60.0, -32.0, -5.0, -42.0], [601606.86, -1530950.07]), ('control #4', [-26.25, -1.472, 0.0, 40.0, 40.0, 40.0], [-3459655.33, -3813601.11]), ('control #5', [24.242, -15.571, 20.0, -45.0, -45.0, -45.0], [499045.1, 3152788.69]), ('boundary #28', [-96.0, 23.0, -96.0, 23.0, 33.0, 45.0], [0.0, 0.0]), ('boundary #29', [5.0, 40.0, 5.0, 40.0, 40.0, 40.0], [0.0, 0.0])], [('control #1', [-7.054, -2.555, 10.0, 52.0, 35.0, 65.0], [-2368752.85, -5460256.95]), ('control #2', [163.654, -41.81, 132.0, 0.0, -18.0, -36.0], [2661851.12, -4920407.16]), ('control #5', [24.242, -15.571, 20.0, -45.0, -45.0, -45.0], [499045.1, 3152788.69]), ('control #7', [-119.961, 31.565, -96.0, 23.0, 33.0, 45.0], [-2250314.51, 1235327.67]), ('control #8', [-25.931, 8.989, 10.0, 52.0, 35.0, 65.0], [-4374379.83, -3619965.56]), ('control #10', [-40.992, -24.699, -60.0, -32.0, -5.0, -42.0], [1816236.48, 739740.04]), ('boundary #28', [-96.0, 23.0, -96.0, 23.0, 33.0, 45.0], [0.0, 0.0]), ('boundary #29', [5.0, 40.0, 5.0, 40.0, 40.0, 40.0], [0.0, 0.0])], [('control #2', [163.654, -41.81, 132.0, 0.0, -18.0, -36.0], [2661851.12, -4920407.16]), ('control #3', [-52.492, -45.412, -60.0, -32.0, -5.0, -42.0], [601606.86, -1530950.07]), ('control #10', [-40.992, -24.699, -60.0, -32.0, -5.0, -42.0], [1816236.48, 739740.04]), ('control #11', [25.244, 44.637, 0.0, 40.0, 40.0, 40.0], [1977746.83, 796959.72]), ('control #12', [-19.298, -12.334, 20.0, -45.0, -45.0, -45.0], [-4597338.48, 2353100.16]), ('control #14', [-128.991, 27.029, -96.0, 23.0, 33.0, 45.0], [-3243071.92, 1029568.66]), ('boundary #28', [-96.0, 23.0, -96.0, 23.0, 33.0, 45.0], [0.0, 0.0]), ('boundary #29', [5.0, 40.0, 5.0, 40.0, 40.0, 40.0], [0.0, 0.0])], [('control #3', [-52.492, -45.412, -60.0, -32.0, -5.0, -42.0], [601606.86, -1530950.07]), ('control #7', [-119.961, 31.565, -96.0, 23.0, 33.0, 45.0], [-2250314.51, 1235327.67]), ('control #14', [-128.991, 27.029, -96.0, 23.0, 33.0, 45.0], [-3243071.92, 1029568.66]), ('control #15', [-24.721, 3.031, 10.0, 52.0, 35.0, 65.0], [-4481724.64, -4195384.79]), ('control #17', [-37.667, -43.269, -60.0, -32.0, -5.0, -42.0], [1818158.31, -1416145.5]), ('control #19', [-7.283, -57.541, 20.0, -45.0, -45.0, -45.0], [-1648585.12, -1661270.39]), ('boundary #28', [-96.0, 23.0, -96.0, 23.0, 33.0, 45.0], [0.0, 0.0]), ('boundary #29', [5.0, 40.0, 5.0, 40.0, 40.0, 40.0], [0.0, 0.0])], [('control #4', [-26.25, -1.472, 0.0, 40.0, 40.0, 40.0], [-3459655.33, -3813601.11]), ('control #7', [-119.961, 31.565, -96.0, 23.0, 33.0, 45.0], [-2250314.51, 1235327.67]), ('control #19', [-7.283, -57.541, 20.0, -45.0, -45.0, -45.0], [-1648585.12, -1661270.39]), ('control #21', [-59.567, 18.585, -96.0, 23.0, 33.0, 45.0], [3917697.15, 316916.45]), ('control #22', [40.214, 4.407, 10.0, 52.0, 35.0, 65.0], [3883538.77, -4321985.71]), ('control #24', [-22.139, -46.424, -60.0, -32.0, -5.0, -42.0], [2976538.67, -1998953.15]), ('boundary #28', [-96.0, 23.0, -96.0, 23.0, 33.0, 45.0], [0.0, 0.0]), ('boundary #29', [5.0, 40.0, 5.0, 40.0, 40.0, 40.0], [0.0, 0.0])]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
control #0	`[2405679.11, -375515.8]`	`[2405679.11, -375515.8]`	Passed
control #1	`[-2368752.85, -5460256.95]`	`[-2368752.85, -5460256.95]`	Passed
control #2	`[2661851.12, -4920407.16]`	`[2661851.12, -4920407.16]`	Passed
control #3	`[601606.86, -1530950.07]`	`[601606.86, -1530950.07]`	Passed
control #4	`[-3459655.33, -3813601.11]`	`[-3459655.33, -3813601.11]`	Passed
control #5	`[499045.1, 3152788.69]`	`[499045.1, 3152788.69]`	Passed
boundary #28	`[0.0, 0.0]`	`[0.0, 0.0]`	Passed
boundary #29	`[0.0, 0.0]`	`[0.0, 0.0]`	Passed

SHA-256 / 360044e7193b0b17a63f8ee68218a668c32b674d83643cd619063ec7bf79b9de

Verification & scope

Stipulated deterministic toy contract on a bounded input domain; results are rounded as stated and no conformance with any published standard or library is claimed. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.

Observations recorded using Python 3.12.14 at 2026-09-29T14:48:17.712753+00:00.

Case digest / fd80fe2bfb1ae76b32faa6bce0b5b0ae9977bfaace2dbbd5f0b9a1c845f59b2f